Topological phase transition and robust pseudospin interface states induced by angular perturbation in 2D topological photonic crystals

In recent years the research about topological photonic structures has been a very attractive topic in nanoscience from both a basic science and a technological point of view. In this work we propose a two-dimensional topological photonic structure, composed of a trivial and a topological photonic crystals, made of dumbbell-shaped dielectric rods. The topological behavior is induced by introducing an angular perturbation in the dumbbell-shaped dielectric rods. We show that this composed structure supports pseudospin interface states at the interface between the trivial and topological crystals. Our numerical results show that a bandgap is opened in the band structure by introducing the angular perturbation in the system, lifting the double degeneracy of the double Dirac cone at the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ point of the Brillouin zone, despite keeping the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_6$$\end{document}C6 symmetry group. A pseudospin topological behavior was observed and analyzed with emphasis on the photonic bands at the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Gamma }$$\end{document}Γ point. We have also investigated the robustness of these pseudospin interface states and, according with our numerical results, we conclude that they are robust against defects, disorder and reflection. Finally, we have shown that the two edge modes present energy flux propagating in opposite directions, which is the photonic analogue of the quantum spin Hall effect.


Scientific Reports
| (2023) 13:850 | https://doi.org/10.1038/s41598-023-27868-x www.nature.com/scientificreports/ candidates for technological applications once the flux of light can be controlled without any significant energetic loss or reflection. In this work we propose a two-dimensional topological photonic crystal with a very specific geometry: it is composed of dumbbell-shaped dielectric rods in a triangular lattice with six rods in the unit cell. We will show later that we can induce a complete bandgap in the system, by introducing an angular perturbation in the rods orientation, having as a consequence the emergence of a topological band gap. The physical system addressed here is of current interest for a large audience because of its potential technological applications. For example, topological photonic crystals may be applied as waveguides, optical filters and may enhance the efficiency of optical fibers. Also, from a basic science point of view, photonic crystals presents photonic analogues of physical phenomena such as quantum Hall effect and quantum spin Hall effect. This work is organized as follows. In Sect. "Photonic crystal" we introduce the system and describe its features. In Sect. "Topological phase transition" we introduce the angular perturbation and investigate the topological behavior of the system. In Sect. "Robust pseudospin interface states" we study the emergence of edge states around the interface between the topological and trivial photonic crystals. We also address the robustness of the interface states against defects and disorder. Finally, in Sect. "Conclusions" we summarize the main results obtained in this work.

Photonic crystal
The photonic structure considered here is a triangular photonic crystal of six"artificial atoms" composed by dumbbell-shaped dielectric rods ( ε = 11.7 ), surrounded by air. The construction of the dumbbell-shaped rods is as follows. We consider three cylindrical rods of radius r = 0.13a , with a = 1 μm being the lattice constant. Then, two of them are shifted by d = ±0.5R from the center of the reference cylindrical rod. Here R = a/3 is the distance between the unit cell center and the center of the reference cylindrical rod. Afterward, we consider the region of the reference cylindrical rod which is not superimposed with the other two shifted cylinders as the dumbbell-shaped dielectric rod (see Fig. 1).
The six dumbbell-shaped rods in the unit cell can be rotated around their respective centers by the orientation angle , as illustrated in Fig. 1. We can observe from Fig. 1 that the angular orientation of the rods allows different configurations for the unit cell. Thus, we can change their angular orientation by introducing an angular perturbation, which may open a bandgap at the Ŵ point. Therefore, we expect that this angular perturbation will lead to a topological phase transition from trivial to non-trivial domain 35,39 . The angular perturbation is introduced in the orientation angle of the rods, so that we can write the orientation angle of the i-th dumbbell-shaped rod as Here, i = 1, 2, ..., 6 is the rod index, φ 0 is the initial unperturbed angle and φ is the angular perturbation introduced in our system. Because of the symmetry of the dumbbell-shaped rod, 0 < φ < π . Here we assume φ 0 = π/4 (see Fig. 2a). We should remark that the initial unperturbed angle φ 0 = π/4 is chosen so that we obtain the band structure without bandgap (see Fig. 2b). This is because φ 0 = π/4 corresponds to a tight-binding model with uniform hopping in the same lattice (for details see Ref. 35 ).
It has been reported in the literature that triangular lattices, with 6 artificial atoms in the unit cell, have two irreducible representations in the C 6 point symmetry group, which are associated with the symmetry of the lattice 40 . This point symmetry group allows the occurrence of a double degenerate Dirac cone at the center of the Brillouin zone 41,42 . Moreover, degenerated bands related to the double degeneracy present pseudospin behavior, which is associated to dipole and quadrupole modes, i.e., p x ( p y ) and d xy (d x 2 −y 2 ) orbitals corresponding to odd (even) parity in the real space, respectively 13 . That irreducible representation allows us to write the pseudospin states as 43-45 , The band structure of the system was obtained through the software COMSOL Multiphysics 46 , which is based in the finite element method (FEM). The TM modes ( E z , H x , H y = 0 ), for the unperturbed case φ = 0 , are shown in Fig. 2b. We can observe from Fig. 2b, due to the C 6 symmetry group, a double degenerated Dirac point at Ŵ point www.nature.com/scientificreports/ of the Brillouin zone. In Fig. 2c and d we plot the profile of the electric field along the z direction ( E z ) and the phase of the electric field, respectively, at the double degenerate Dirac point with ωa/(2πc) = 0.5575 . The pseudospin sates that are related to dipole modes correspond to p x and p y orbitals, while the pseudospin sates that are related to quadrupole modes correspond to d xy and d x 2 −y 2 orbitals. We can observe from Fig. 2d that the electric field presents two well defined phases, 0 and π , with an abrupt transition between them. In the next section we will show the procedure to lift the double degeneracy of those states and obtain a complete photonic bandgap. The magnetic field associated to pseudospin up and down states is given by the Faraday relation. For example, for the p orbital we have H ± = −(i/µ 0 ω)∇ × [(p x ± ip y )ẑ] . Since the p x and p y modes are connected to each other by a π/2 rotation, we can write the magnetic field as (for more details see the supplemental material of Ref. 13 The term iŷ corresponds to a ∓π/2 phase shift of the x and y components of the magnetic field, leading to a circular polarization. This circular polarization corresponds to the angular momentum of the electric field. The pseudospin state can be identified according to the circular polarization of the magnetic field, e.g., an anticlockwise circular polarization is associated to the pseudospin up and a clockwise circular polarization is associated to the peudospin down 47 . This is illustrated in Fig. 3, i.e., the pseudospin up is represented by an anticlockwise circular polarization of the magnetic field, while a pseudospin down is represented by a clockwise circular polarization of the magnetic field.

Topological phase transition
In order to lift the double Dirac cone and open a complete photonic bandgap, we induce an angular perturbation on the range −π/2 ≤ φ ≤ π/2 at the Ŵ point, as it is shown in Fig. 4. It should be observed that for the unperturbed case ( φ = 0 ) we have four degenerate bands resulting in the double degenerate Dirac cone. On the other hand, for the perturbed case φ = 0 , the perturbation lift the double degeneracy and a bandgap is opened, despite keeping the C 6 symmetry group. We can observe from Fig. 4 that as |φ| increases from 0, the bandgap width monotonically increases, reaching its maximum at |φ| = π/4 . On the contrary, as |φ| increases from π/4 , www.nature.com/scientificreports/ the bandgap width monotonically decreases, until the double Dirac cone is recovered at |φ| = π/2 . This is a consequence, as mentioned before, that the perturbation has a period of π. It is clear from Fig. 4 that φ opens a bandgap at the Ŵ point for both positive and negative angular perturbation. However, those bandgaps have different topological behavior. Therefore, we need to investigate their topological invariant, the Chern number, for a better understanding of their bulk and edge modes. Thus, in order to study the topological behavior close to the Ŵ point, we write an effective Hamiltonian around it by using the k.p perturbed model. The Chern number of the bandgap is then given by 48-49 , Here B is the diagonal term of the effective Hamiltonian close to the Ŵ point, which is obtained from secondorder correction and is essentially negative 13 . Also, M = (ε d − ε p )/2 , where ε d and ε p are the eigenmodes of orbit d and orbit p, respectively 16 . The eigenmode ε p is related to the doubly degenerate dipole states of p ± , while ε d is related to the doubly degenerate quadrupole states of d ± 48 . If ε p < ε d then M > 0 , hence C ± = 0 and the photonic crystal is topologically trivial. However, if ε p > ε d then M < 0 and C ± = ±1 and the photonic crystal is in a topological phase. So, the inversion of the bands between the degenerated modes at the Ŵ point promotes the topological phase transition 31 . In fact, the inversion of the bands takes place at φ = 0 as we can see in Fig. 5. The phase with φ < 0 corresponds to C ± = 0 , while the phase with φ > 0 corresponds to C ± = ±1 (see Fig. 4).
Let us illustrate the inversion of the bands and its relation with the topological phase. We consider the inversion of the bands that takes place between φ = −π/4 and φ = π/4 for the degenerated bands, as shown in Fig. 5. It is possible to notice from Fig. 5a that for the negative perturbation case ( φ < 0 ) the frequency of the dipole modes is lower than the frequency of the quadrupole modes, while for the positive perturbation case ( φ > 0 ) the frequency of the dipole modes is higher than the frequency of the quadrupole modes. Therefore, for −π/4 ≤ φ < 0 we obtain C ± = 0 that corresponds to a trivial photonic crystal. On the other hand, for the case 0 < φ ≤ π/4 we obtain C ± = ±1 that corresponds to a topological photonic crystal, as expected. In addition to that, Fig. 5b shows the phase of the electric field for the cases of φ = −π/4 and φ = π/4 . It is possible to notice a phase shift of π , when the topological phase transition takes place, in agreement with the literature 34 .
The band structures for the trivial and topological photonic crystals, with angular perturbation φ = 0 , are shown in Fig. 6. The perturbation opens a gap between ωa/(2πc) = 0.5462 and ωa/(2πc) = 0.5657 , for the trivial photonic crystal case, and between ωa/(2πc) = 0.5453 and ωa/(2πc) = 0.5667 , for the topological photonic crystal case, corresponding to gap-midgap ratios 50 of �ω/ω m = 0.0350 and �ω/ω m = 0.0385 , respectively.  www.nature.com/scientificreports/ Once the bandgap is opened, edge states can appear inside the gap and these states may present very interesting features as we will see in the next section.

Robust pseudospin interface states
The perturbed system presents both the topological and trivial bandgaps associated to the angular perturbation φ of the dumbbell-shaped rods. It is known from the literature that the edge-bulk correspondence guarantees that if we build a slab composed of two photonic crystals, with different topological invariants, i.e., Chern numbers, robust edge modes, localized around the interfaces between the photonic crystals, emerge 45,[51][52][53] . Once the edge modes are localized at the interfaces, the corresponding edge states are called interface states. The interface states are topologically protected and are robust against defects, disorder and allow transmission without any reflection, with no significant energetic loss 28,54,55 . In order to investigate the emergence of the interface states in our For −π/4 ≤ φ < 0 the frequency of the dipole modes is lower than the frequency of the quadrupole modes, corresponding to the trivial case (left panels). For 0 < φ ≤ π/4 the frequency of the dipole modes is higher than the frequency of the quadrupole modes, corresponding to the topological case (right panels). (b) Phase of the electric field E z . We can observe that the topological phase transition is also characterized by a phase shift of π in the electric field. (We used COMSOL Multiphysics v.6.0. www. comsol. com. sofware to create the image). www.nature.com/scientificreports/ system, we build a supercell with 15 trivial unit cells ( φ = −π/4 ) and 15 topological unit cells ( φ = π/4 ), with an interface between them, and then we calculate the projected band structure along the Ŵ → M direction, as we can see in Fig. 7. It is possible to observe two topological interface modes inside the gap, which present a mirror symmetry around the Ŵ point, i.e. ωa/(2πc)(−k) = −ωa/(2πc)(k) . From Fig. 7b we can observe that they cross each other right at the Ŵ point, which means that the two topological interface modes are degenerated at the Ŵ point. Moreover, taking into account the group velocity v g = ∇ k ω , we conclude that those modes are traveling in different directions. In Fig. 7c and d are shown the spacial distributions of E z , along the supercell, the time-averaged Poynting vector ( S = Re[E × B * ]/2 ), and the phase of E z , around the interface, for ωa/(2πc) = 0.5532 and ωa/(2πc) = 0.5596 , respectively, with k = k xx = 0.015 k 0x and k 0 = 4π/( √ 3a) . It is easy to observe that the modes are localized around the interface, so that the intensity of the electric field is reduced in the bulk (we will quantify this behavior later). Furthermore, by comparing their Poynting vectors, it is clear that the interface states present energy flux in different directions (see the arrows in Figs. 7c and d). Moreover, we can identify that the mode with ωa/(2πc) = 0.5532 presents a clockwise polarization, while the mode with ωa/(2πc) = 0.5596 presents an anticlockwise polarization, which correspond to a pseudsopin down and up, respectively. So, we can associate a pseudospin number to each interface mode ( ±1/2 ) according to the propagation direction and energy flux 56 . The pseudospin-up ( +1/2 ) is associated to the interface state with group velocity and energy flux from the left to the right, while the pseudospin-down ( −1/2 ) is associated to the interface state with group velocity and energy flux from the right to the left 14 (see Fig. 7b). Fig. 7c and Fig. 7d show the spatial distribution of E z for pseudospin-down and -up, respectively. The electromagnetic energy propagates at the interface in opposite directions according to those two pseudospins states ( ±1/2 ). This behavior is analogous to the quantum spin Hall effect (QSHE). Therefore, we may conclude that the structure investigated here presents the photonic spin Hall effect (PSHE) [57][58][59][60] .
Let us now investigate the robustness of the pseudospin interface states against disorder and defects, as well as, their localization at the interface [61][62][63][64] . To do so, we build a (25, 20 √ 3/2)a slab with an interface between the trivial ( φ = −π/4 ) and topological ( φ = π/4 ) photonic crystals. We consider a source of propagating waves by using a"port on" on the left and a detector by using a"port off " on the right, as it is depicted in Fig. 8a. Then, we plot the normalized electric field profile defined as E N = |E z | 2 for ωa/(2πc) = 0.5462 , as it is shown in Fig. 8b. We also considered different defects at the interface: (i) a cavity by removing a rod (see Fig. 8c), (ii) a larger dumbbell-shaped rod of size d 0 = 0.6R (the bigger one in Fig. 8e), (iii) an Al dumbbell-shaped rod (see Fig. 8g), (iv) a Z interface (see Fig. 8i), (v) an Omega interface (see Fig. 8k) and (vi) a double horseshoe interface (see Fig. 8m). The normalized electric field profiles E N corresponding to these defects, for ωa/(2πc) = 0.5462 , are shown in Figs. 8d, f, h, j, l, and n, respectively. Besides the normalized electric field profiles, the time-averaged Poynting vectors are also illustrated in Fig. 8 (see the arrows in the zoom windows). We can observe from Fig.  b, d, f, h and j, that the electric field survives and the energy flux remains unchanged along the interface for all defects considered here. Notice that the Poynting vectors present small changes only around the defects or around the corners of the interfaces, but their global behavior and direction (see the big arrows) remain unchanged for all defects considered here.
It is important to realize from Fig. 7a that the supercell present an interface composed of a complete unit cell alternating with a broken unit cell. The results obtained from this interface are completely analogous to the results obtained from an interface purely composed of a complete unit cell, or an interface purely composed of a broken unit cell. Thus, the interface states are going to emerge inside the bandgap and these states present a pseudospin behavior no matter is the interface's shape. In fact, this is an expected result, once the emergence of the interface modes is fully guaranteed by the Bulk-Edge correspondence. It means that the only request for the emergence of the edge modes around the interface is the nonzero difference between Chern numbers from the crystals that compose the supercell in Fig. 7a and the sample in Fig. 8.
The qualitative information from Fig. 8 is corroborated by the quantitative information provided by the transmittance of the system 22,65,66 . In order to confirm that statement, the transmittance between "port on"and "port off ",corresponding to the slabs schematized in Fig. 8, are plotted in Fig. 9. We can observe from Fig. 9 that the transmittance peak practically does not change for the case without defect, cavity defect, larger rod defect and Al rod defect. We could expect some losses when we introduce the Al defect, once it is made by a metallic material, but those losses are much smaller than the transmittance and we can neglect them. On the other hand, for the Omega interface, Z interface and double horseshoe interface, the transmittance peak goes around 1.0, 0.7 and 0.8, respectively. Those cases present different behaviors because they are built by bending the orientation of parts of the original interface. Note that for: (i) the Z interface case light faces two direction changes; (ii) the Omega interface case light faces six direction changes; and (iii) the double horseshoe interface case light faces six direction changes with larger angles. Therefore, once those interface cases are not small and localized defects, it is expected that its transmittance be a little different from the previous cases. As an overall conclusion, our results show with no doubt that the transmittance peaks survive even in the presence of the complex defects considered here. Thus, it is shown that the pseudospin interface states are robust against defects and disorders introduced into the system.
Before concluding, let us analyze from a quantitative point of view the localization of the pseudospin states around the interface. This localization was illustrated from a qualitative point of view in Fig. 8. In order to quantify the localization, we calculate E N /E 0 point-by-point on a line perpendicular to the interface (see Fig. 8). Here E 0 = k e e/a 2 , where k e is the Coulomb constant, a is the lattice constant and e is the modulus of the elementary electric charge. Figs. 10a-g show the intensity of E N /E 0 , for ωa/(πc) = 0.5462 , taking into account the interface: (i) without defect, (ii) with cavity defect, (iii) with a larger rod, (iv) with an Al rod, (v) with a Z interface, (vi) with an Omega interface and (vii) with a double horseshoe interface, respectively. Analyzing Fig. 10 it is possible to observe that the electric field is localized around the interfaces, inclusive in the presence of defects and   www.nature.com/scientificreports/ also for the cases of the Z interface, the Omega interface and the double horseshoe interface. The electric field in the bulk, far from the interface, is very low in both the topological and trivial photonic crystals. In particular, for the cases of the Z interface, the Omega interface and the double horseshoe interface, we plot E N /E 0 , also for ωa/(πc) = 0.5462 , on two lines: (i) line 1 located before the corners of the interface and (ii) line 2 located after the corners of the interface. Those calculations are illustrated in Fig. 10e-g. Once again it is confirmed the localization of the electric field around the interface, no matter in which part of the interface the calculation is performed. Therefore, we can conclude that the system proposed in this work presents a typical topological insulator behavior, i.e., it supports robust pseudospin states, allowing propagation at the border (interface), but with no propagation in the bulk [67][68][69][70] .

Conclusions
In summary, we have proposed a 2D topological photonic system, composed of dumbbell-shaped dielectric rods in a triangular lattice with six rods in the unit cell. The band structure of the system was obtained through the software COMSOL Multiphysics 46 , which is based in the finite element method (FEM). We have found that a complete bandgap is opened in the system by introducing an angular perturbation φ in the rods orientation. As a consequence of the perturbation, two interface states emerge between the topological and trivial photonic crystal, which are localized around the interface and topologically protected. Our results show that the interface states present energy flux propagating in opposite directions. In addition to that, they have opposite winding of the electromagnetic field's phase and Poynting vector, so that we can associate a pseudospin number to each interface mode (±1/2) , now called pseudospin interface states according to the rotation direction of Poynting vector. This behavior is analogous to the quantum spin Hall effect (QSHE). Accordingly, the structure investigated here presents the photonic spin Hall effect (PSHE). Furthermore, topological protection makes the pseudospin interface states robust against defects and disorder. This is well illustrated trough the calculation of the transmittance of the system for: (i) a cavity defect, (ii) a larger rod defect, (iii) an Al rod defect, (iv) a Z interface, (v) an Omega interface and (vi) a double horseshoe interface. For all defect cases considered here the transmittance is mildly affected, which means that the system proposed is an excellent candidate for technological applications, once the flux of light can be controlled without any significant energetic loss or reflection. Finally, from the calculation of the electric field intensity far from the interface, we can conclude that the system proposed in this work presents a typical topological insulator behavior, i.e. it supports robust pseudospin states, allowing propagation at the border (interface), but with no propagation in the bulk. As a concluding remark, it is worth to emphasize that the angular perturbation introduced in this work has lifted the double degeneracy, of the double Dirac cone at the Ŵ point of the Brillouin zone, despite keeping the C 6 symmetry group. Certainly the 2D topological photonic system proposed in this work can be realized experimentally, and we hope that experimentalists are encouraged to investigate it.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.